A Compact Representation for Scanned 3D Objects

Bing Wang and Holly RushmeierDepartment of Computer Science

Yale University{bing.wang, holly.rushmeier}@yale.edu

Abstract

We consider the representation of densely sampledscanned 3D objects. Scanning physical objects can be anefficient way of obtaining natural looking input for com-puter graphics image generation. Scanned meshes howeverrequire large amounts of storage. For applications suchas computer graphics, preserving the look of objects ratherthan precise measurements is critical. We seek a represen-tation that preserves characteristic features at both low andhigh spatial frequencies. Our proposed representation con-sists of a simplified base mesh and characteristic detailedpatches. The detailed patches contain a spatially densesampling of mean curvature on a small region on the objectsurface. A large initial set of patches covering the objectis segmented into sets of patches using k-means clustering.Each cluster of patches is then compactly represented withGaussian distributions. We present comparisons of a seriesof original scanned objects and the objects reconstructedfrom the compact representation.

1. Introduction

The widespread availability of 3D scanners makes it pos-sible to get densely sampled models of physical objects.The scanned models, usually in the form of triangle meshes,are very precise and detailed representations of objects.Such representations are highly useful in applications suchas the documentation of cultural heritage.

However, for other applications such as computer graph-ics, it is more important to preserve the characteristic fea-tures of objects than the precise measurements. From a par-ticular scanned model, it is useful to generate a variety ofnovel models which look similar but different in order topopulate a synthetic scene. Because the densely sampledtriangle mesh doesn’t directly encode the characteristic fea-tures of an object, it can not be used directly in authoringsimilar objects.

In this paper we present a compact representation ofdensely sampled model which consists of a much simpli-

fied base mesh and representative patches. This is based onthe observation that local areas of objects often experiencestatistical similarity and structural correlation. For example,rocks are bumpy all around the surface and seashells havethe characteristic spiral curves. We cluster into groups thedensely sampled local patches using a k-means method anduse Gaussian distributions to model each group of patches.This statistical model of local patches not only saves storageand network bandwidth, but also provides us with a methodto generate similar models.

The main contribution of the paper includes: (1) a sta-tistical model of local patches; (2) curvature-domain localshape descriptor; (3) a compact mesh representation basedon statistical patch model; and (4) approaches of automati-cally authoring similar models.

The rest of the paper is organized as follows: after reviewthe related literature in section 2, we introduce the statisticalpatch model in section 3 and the compact mesh representa-tion in section 4. Results are shown in section 5 and futurework is discussed in section 6.

2. Related work

The work presented here builds on several areas of ge-ometric modeling – modeling using 3D scan data, multi-resolution meshes, detecting and exploiting redundancy inmodels, and geometric texture transfer. We briefly reviewthis previous work, and highlight how the problem and so-lution we consider differs from it.

Parts of existing models, including models produced by3D scanning, can be used as input for new models [7]. How-ever rather than using entire “chunks” of existing models,we hope to reuse characteristic shapes of objects, particu-larly natural objects such as rocks. Characteristic shapesare associated with materials [1].

In computer graphics, procedural modeling has domi-nated the creation of models of natural objects. For exam-ple, there have been efforts at finding mathematical descrip-tions of the basic shapes of natural objects such as shells [6].To model natural irregularities in shape procedures such asPerlin noise [21] are added to geometries representing nat-

1793 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops978-1-4244-4441-0/09/$25.00 ©2009 IEEE

ural shapes such as terrains defined as height fields [20].Natural looking effects have relied on manual tuning of pa-rameters, or using general image statistics [22]. Followingone aspect of the procedural approach, we seek to definemodels in terms of an overall form that is modulated byhigh spatial frequency details.

Multiresolution representation for geometric models is ahighly developed area for efficient shape editing and ren-dering, e.g. [9]. The representations consist of multiplemeshes capturing geometric features at different frequencybands. Since eliminating redundancy is not a goal of therepresentation, precise detailed information is recorded atmultiple levels. The mesh at a particular level can be ob-tained by adding the recorded details into the mesh at theadjacent lower level. In our work we concern with separat-ing the basic form of an object and the shape details that arecharacteristic of the material forming the object, so we donot consider these multi-level representations but simply atwo level representation.

Since we seek a compact representation, we look for re-dundancy in models. One approach to this is symmetryanalysis of shapes which aims to find spatial invariant ofgeometry elements under 3D transfer, rotation and scaling,e.g. [16]. This kind of symmetry is widely presented insome objects such as buildings. Other objects like rocksdon’t necessarily have the “perfect” symmetry. The work inthis paper doesn’t rely on repetitions but rather on similari-ties in a statistical sense.

To find groups of similar, but not identical features, weconsider clustering. Clustering has been applied long beforefor understanding images. Clustering textons into multiplegroups is usually one of the basic steps for image segmen-tation and texture classification, e.g. [23]. For regular tex-tures where textons are tiled in structure, researchers haveproposed approaches to automatically identify the textonsand extract their spatial organization, e.g. [14]. These ap-proaches are similar to those of shape symmetry analysis,and don’t extend to statistical textures directly. For the re-sults of clustering, we will not try to find a specific regularstructure.

Given a description of “typical” high spatial frequencyfeatures, a natural mechanism for applying these featuresto a basic form is texture transfer. Many methods havebeen developed recently for the transfer of geometric tex-ture. These may be divided into two classes – methods thattransfer a different material or different ornamental featuresto a base form, and methods that enhance the surface vari-ations of the basic object. An example of a detailed ad-ditional material is adding a volumetric layer to a surfaceto represent fur [10]. Examples of ornamental features in-clude wire meshes, basket weaves, or patterns of repeatedsmall decals that are represented as meshes on a flat surfacethat are deformed and repeated to cover a shape [5, 24]. Ad-

ditional material or ornamental features are generally mod-eled separately, rather than being extracted from an existingmodel.

The second category of geometric texture transfer, thatenhances the surface variations of the base form rather thanadding a new layer of material, is relevant to the problemwe address. In this case, methods have been developedthat attempt to extract detail from basic shape of one ob-ject, and then apply detail to a new shape. Bhat et al.[2]synthesize geometric textures using voxel definitions of theobject and fine scale geometry. A vector field on the sur-face is used to guide the synthesis on the base surface. Laiet al.[12] transfer using geometry images rather than volu-metric structures. In both cases the transferred geometry isfound from an existing object by defining a smoothed andsimplified base object. The geometric height field or geo-metric structure then depends on the specific simplificationthat is made. In our method we attempt to find a detailedsurface description that is independent of arbitrarily speci-fying a surface depth to define the difference between smalland large scale features.

In the general field of texture transfer it has been rec-ognized that the texture should be correlated to large scalegeometric features. Texture transfer that uses analysis ofunderlying geometry has been proposed in particular to cap-ture the look of weathered objects [15, 17]. Accounting forpattern variations with large scale variations is importantin weathered appearance since it depends on the actions ofnatural phenomena. Similarly, to capture the fine scale geo-metric variations of natural objects we will account for theunderlying large scale features.

Many different types of surface maps have been pro-posed to model surface detail, including texture, bump, dis-placement and normal maps [19]. Maps are an effective wayto represent fine scale features in a compact form. For ex-ample, de Toledo et al. consider using height fields as a ge-ometry texture map over an entire object to store mesostruc-ture [3]. However, synthesizing fine scale geometry as dis-places or normals requires careful and time consuming at-tention to continuity to avoid obvious artificial discontinu-ities in surfaces.

A way to avoid the problems of arbitrarily defining adepth layer for details and the discontinuities in synthesisof displacement or normal maps, is to characterize detailsin the curvature domain. Eigensatz introduced the idea ofmodeling in the curvature domain [4]. Working with cur-vature captures the appearance of the object without firstdefining a base mesh for offset, and without compensatingfor a particular coordinate system. Working with a higherorder quantity such as curvature and then synthesizing thesurface avoids obvious surface discontinuities.

Based on this assessment of previous work, in the nextsections we develop our approach for compactly represent-

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ing surfaces. By working in the curvature domain we invertthe usual process of simplifying a surface and then definingdetails by differencing the detailed and simplified meshes.In the next two sections we describe how we first analyzethe curvature variations in small local areas (i.e. patches) ona mesh and define a small set of patch descriptions, and thendefine a compact mesh representation designed to make useof these patch descriptions.

3. A Statistical Model for Local Patches

Our analysis of an object’s shape begins by consideringthe characteristics of small areas distributed over the objectsurface. We can consider characteristics without referenceto a base mesh or to the global coordinate system by work-ing in the curvature domain. Similar local areas are oftenobserved at different surface locations of an object. We in-troduce a model in this section to express this similarity sta-tistically using clustering and Gaussian distributions.

3.1. Shape descriptor for local patches

We consider a collection of vertices V, edges E and facesF that form a triangle mesh M = {V, E, F}, which ap-proximates a smooth 2-manifold. Let NV = |V | denote thenumber of vertices, let pi denote the i-th vertex and let xi,ni and Hi denote the position, normal and mean curvatureof pi, respectively.

Given a vertex pi on M and a small radius r, there is aneighborhood of pi which consists of all the points on Mwith a geodesic distance to pi less than r. Such a neighbor-hood is called a local patch of pi, denoted by Pi.

A shape descriptor is computed for each local patch. Theshape descriptor should have the power to reconstruct thepatch. Mean curvature is used because of its character ofbeing intrinsic (independent of the global coordinate sys-tem) and because no reference base mesh is required ( as isrequired to define a height field). Mean curvatures at ver-tices are computed using the mesh Laplacian with cotan-gent weights [18] as shown in equation 1, where Ai is one-third of the 1-ring neighborhood area of pi, N1(i) is the setof 1-ring neighborhood vertices of pi, and αij and βij arethe opposite angles of edge (pi, pj), as shown in Figure 1.Mean curvatures at other surface points are computed bybilinearly interpolating vertex curvatures using barycentriccoordinates. We uniformly sample the mean curvatures onthe local patch and form a vector of mean curvatures as theshape descriptor for this patch.

2Hini =1

2Ai

∑pj∈N1(i)

(cotαij + cotβij)(xi − xj) (1)

Local parameterization is used to efficiently compute thelocal patch and the spatially regular sampling positions. For

pi pjpi pj

ij

ij

Figure 1. 1-ring neighborhood of a vertex

a particular vertex pi, we first parameterize its neighbor-hood onto a 2D domain, with pi mapped to the origin, us-ing LSCM (Least Squares Conformal Maps) [13]. The pa-rameterization produces, for each surface point within theneighborhood, a corresponding (u, v) parameterization co-ordinate. By approximating the geodesic distance on thesurface with the Euclidean distance on the parameterizationdomain, the local patch Pi is then computed as the set ofsurface points whose parameterization coordinates have theproperty as u2 + v2 < r2, that is, the set of points whosecorresponding points on the parameterization domain locatewithin the circle centering at the origin and with a radius ofr.

The sampling process for computing the shape descrip-tor is also done in the parameterization domain. Given asampling distance d, all the sampling positions are found asthe lattice points in the parameterization domain as shownin Figure 2(d). By composing the mean curvatures sampledat the corresponding surface points into a vector, we get theshape descriptor Si = (Hij) with 1 ≤ j ≤ Ns, where Hij

is the mean curvature sampled at the j-th sampling point ofthe i-th patch and Ns is the number of sampling positions.

In practice, the parameterization domain is rotated be-fore the computation of shape descriptor such that the mostsignificant direction always coincides with the X-axis, asshown in Figure 2(c). The most significant direction isdefined as the direction which has the maximum absolutevalue of average mean curvatures. An orientational his-togram is used to compute this direction and only the meancurvatures at vertices are considered to accelerate the pro-cess.

3.2. K-means clustering and Gaussian distribution

By sampling at different local areas along the object sur-face, we get multiple local patches and their shape descrip-tors. Let NP denote the number of local patches. Those lo-cal patches may have distinct shapes. As shown in Figure 3,for example, patches along the edges of the object and thoseat the flatter area are quite different. The difference is mea-sured with Euclidean distance in the high dimensional shapedescriptor space. A k-means clustering method (Lloyd’s al-

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(a) Local patch in 3D. Meancurvatures are visualized withred for large and blue for small.

(b) Local patch in 2D

(c) Reoriented local patch in2D

d

rr

(d) Regular sampling

Figure 2. Compute shape descriptor for a local patch.

(a) Clustering result for stoneA. (b) Clustering result for stoneC.

(c) Clustering result for stoneL. (d) Clustering result forWooden House.

Figure 3. Clustering results. Different color stands for local areasof different clusters.

gorithm, as described in [11]) is applied to group the NP

shape descriptors into K categories Gi, 1 ≤ i ≤ K , with|Gi| = NGi .

The shape descriptors in each group Gi form a cloud ofNGi points in a Ns-dimensional space. We use a 1D Gaus-sian model to represent the distribution of the point cloud ineach dimension, which produces a compact representationwith the sacrifice of losing the correlation information be-tween different dimensions. The Gaussian distribution forthe j-th dimension of i-th group is shown in equation 2.

Gaussij(x) =1

σij

√2π

exp(−(x− μij)2

2σ2ij

) (2)

The mean μij is computed as

μij =

∑k|Sk∈Gi

Hkj

NGi

. (3)

The standard deviation σij is computed as

σij =

√√√√ 1NGi − 1

∑k|Sk∈Gi

(Hkj − μij)2. (4)

4. A compact mesh representation

The densely sampled triangle mesh can be compactlyrepresented using a base mesh which captures the low fre-quency features and a group of Gaussian distributions whichcapture the high frequency features. Rather than using ex-isting simplification methods (e.g. [8]) that see to maintaindimensional accuracy, we define a base mesh that is com-patible with using Gaussian distributions of patch curvaturedescribed in section 3.

4.1. Encoding the base mesh

In order to record the low frequency and large scale fea-tures of the object, we first simplify the input dense meshM to a base mesh Mb. The simplification is done in twosteps, choosing the retained vertices and reconstructing thetriangle mesh from the retained vertices.

We choose retained vertices from the original mesh M .The selection is constrained by two factors: (1) the verticesare uniformly sampled along the surface such that each hasa neighborhood approximating the predefined patch size,that is, the distance between adjacent vertices should beabout twice the patch radius r so there will be no largeoverlaps or gaps between patches; (2) the retained verticesshould capture the geometric features as well as possible.Mean curvatures are used as guidance for vertex selection,and the vertex with larger absolute value of mean curvaturewill be selected with higher priority than others, since thesevertices represent areas with distinctive features.

A triangle mesh is then built up from the retained ver-tices. We first flood-fill the surface of the mesh M from allof the retained vertices and get a partition of the mesh. Thenwe compute the dual graph of the partition to get a polygonmesh M ′ and M ′ is then converted to a triangle mesh Mb byfurther subdividing each non-triangle polygon into multipletriangles.

The resulting base mesh is formed by a much smaller setof vertices that all lie on the original mesh.

4.2. Encoding the details

The local shapes of the dense mesh is recorded usingstatistical patch models introduced in section 3. For each

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retained vertex, we find the local patch on the dense meshM with a predefined size r. We compute a shape descriptorfor each neighborhood. We then perform patch analysis asshown in section 3 on those shape descriptors. Finally weget a Gaussian distribution parameters table of K rows andNs columns. For each vertex in the base mesh Mb, we keepan index to the table rows to record the type of the localpatch, and a scalar value to record the rotation angle forreorienting the local patch.

4.3. Decoding

We can reconstruct a dense mesh from the compact rep-resentation in four steps: (1) tessellate the base mesh intohigher resolution; (2) for each original vertex on the basemesh, compute a local patch shape descriptor using therecorded statistical model and reorientation angle; (3) re-sample the shape descriptors so all vertices of the tessel-lated mesh have a mean curvature defined; (4) reconstructthe mesh from vertex mean curvatures.

The reconstruction is formulated as an energy minimiza-tion problem

arg minX(w2H‖L(X)−H(X)‖2 + w2

p‖Xb−X0b ‖2), (5)

where X denotes the current vertex positions, H(X) de-notes the vertex mean curvature normals, L(X) denotes themesh Laplacian with cotangent weights, Xb denotes the cur-rent positions of vertices of the base mesh and X0

b denotesthe original positions of those vertices. wH and wp denotethe weight of mean curvature constraints and the weight ofpositional constraints. That is, we need to solve in leastsquares sense the following system

(wHLwpI

)X =

(wHH(X)

wpX0b

). (6)

The system is nonlinear because both L(X) and H(X) arenonlinearly dependent on X . We solve it by iteratively up-dating L and H using current X and then updating X bysolving a linear system.

5. Results

We demonstrate in this section results of using statisticalpatch models to compactly represent dense triangle meshesand geometric detail transfer based on the compact repre-sentation. All the experiments were done on a 3.0 GHzPentium IV PC with 3GB memory. Predefined values ofpatch size r = 5.0mm and sampling distance d = 0.5mmare used for all models. Table 1 shows the times of comput-ing the compact representations and reconstructing meshesfrom them. Please note that we deliberately choose flatshading to render all the meshes to better show how the ge-ometry of the meshes are represented.

5.1. Clustering

Clustering is used to find statistical similarities and char-acteristic features of objects. Figure 3 shows the clusteringresults of several objects, each having been clustered into4-6 groups. Small number of clusters are used for better vi-sualization. Ridges, valleys and flat areas are well separatedinto different groups. Because each local patch is reorientedto the most significant direction, similar patches with differ-ent 3D poses are still grouped together.

5.2. Compact representation

A dense triangle mesh is represented with a much simpli-fied base mesh and representative local patch models. Thesize of the compact representation is therefore determinedby two factors: (1) the size of the base mesh, which de-pends on the patch size r; (2) the size of the table of patchmodels, which depends on the number of clusters K andthe sampling distance d. Since both r and d are predefinedin the experiments, only K influences the final size of therepresentation. K for fractal surfaces such as bumpy rockscan be quite small for an acceptable reconstruction. In ourexperiments for various stones, we always set K to 4. Forother objects like the Buddha shown in Figure 4(j) - 4(l),a larger K has to be used to account for its many featuressuch as the nose, eyes and hairs, etc., which are differentand unique.

5.3. Reconstruction

Figure 4 shows results of reconstructing dense meshesfrom compact representations. It can be seen that the com-pact representations add detail to the simplified meshes.However, compared to original meshes, the reconstructedones are still missing high frequency information and thuslook more smooth. We believe that is because the Gaus-sian distribution model is not adequate to accurately recorddetail geometry. We will exploit more advanced statisticalmodels in future.

5.4. Detail transfer

The Gaussian distributions represent characteristicpatches of objects, and can be directly used to transfer de-tail geometry from one object to another. In Figure 5, weshow examples of transferring detail geometry of stoneAas shown in Figure 4(a) to several objects. The transfer isguided by the correlation between the based meshes. Thatis, for each local patch of the target base mesh, we computea shape descriptor and then identify its cluster by finding itsnearest neighbor in the space of the shape descriptors fromthe source mesh. In this way, we can get Gaussian distri-bution models from the source mesh for each local patch ofthe target.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 4. Reconstructions from compact representations. Left column: original meshes; center column: simplified meshes; right column:reconstructed meshes. First row: StoneA; second row: StoneB; third row: StoneC; last row: Buddha.

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M size of M size of Mc Tsimp Tsd Tstat Tcompcurv Trecon

StoneA 2.23 0.15 27 67 6 89 23StoneB 1.90 0.12 32 40 1 62 15StoneC 0.95 0.07 16 19 1 34 7Buddha 3.81 0.59 108 30 15 49 10

Table 1. Timing results and sizes of representations from our experiments. Time is in seconds. Size is in MBytes. Mc stands for thecompact representation which includes both the base mesh and the statistical patch models. Tsimp, Tsd and Tstat stand for the timecomputing base mesh, shape descriptors and statistical patch models, respectively. Tcompcurv and Trecon stand for the time of recoveringcurvatures from statistical patch models and reconstructing mesh from curvatures.

(a) Toy model. (b) Mechanics model. (c) Wooden House model.

(d) Toy model with detail transferred. (e) Mechanics model with detail transferred. (f) Wooden House model with detail trans-ferred.

Figure 5. Geometry detail transfer based on statistical patch models. First row: original meshes. Second row: meshes with transferreddetail.

6. Future work

We have described a method for compactly representing3D scanned objects that decomposes the object into a baseform and a small set of detailed patches. This representationis useful for representing natural objects for graphics ap-plications in which capturing characteristic features, ratherthan precise dimensions, is desirable. The proposed statis-tical model can be used to transfer geometric features andthus involve in authoring novel objects, which makes it dif-ferent than traditional mesh compression methods and con-ventional detail-enriching rendering techniques based onsurface maps.

The main value of this work is that it introduces a num-ber of promising avenues for future research in the area ofusing 3D scanned data in authoring systems for objects incomputer graphics. These directions include:

• Automatically determining the optimal patch size andnumber of clusters for the surface details. An obviousstarting point for this is a brute force approach that triesmultiple scales and selects the scale that best exploitsredundancy.

• Developing a statistical patch description that accountsfor correlations across dimensions.

• Finding the best method for determining the matching

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of detailed patch to mesh vertex for newly authoredshapes.

• Finding a method for statistically characterizing thebase shape as well as the detailed features by analyzinga collection of similar scanned objects.

• Finding a method that includes variations of re-flectance (i.e. spatial varying bidirectional reflectancedistribution functions) that are correlated to geometricdetail variations.

The success of research along these lines will facilitatethe efficient modeling of complex natural scenes by elim-inating manual parameter tuning and storing the results ofmodeling in a compact form.

Acknowledgements This material is based upon work sup-ported by the National Science Foundation under Grant No.0528204.

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